5. COSMOLOGICAL CONSTANT

The strongest evidence for a positive
comes from
high-redshift SNe Ia, and independently from a combination of
observations indicating that
m ~ 0.3 together with CMB
data indicating that the universe is nearly flat. We will discuss
these observations in the next section. Here we will start by looking
at other constraints on .

The cosmological effects of a cosmological constant are not difficult
to understand
(Felton & Isaacman 1986;
Lahav et al. 1991;
Carroll, Press, & Turner
1992).
In the early universe, the density of energy
and matter is far more important than the
term on the
r.h.s. of the Friedmann equation. But the average matter density
decreases as the universe expands, and at a rather low redshift
(z
~ 0.2 for m = 0.3,
= 0.7) the
term
finally becomes dominant. Around this redshift, the
term
almost balances the attraction of the matter, and the scale factor a (1 + z)-1
increases very slowly, although it ultimately
starts increasing exponentially as the universe starts inflating under
the influence of the increasingly dominant
term. The
existence of a period during which expansion slows while the clock
runs explains why t0 can be greater than for
= 0, but this
also shows that there is an increased likelihood of finding galaxies
in the redshift interval when the expansion slowed, and a
correspondingly increased opportunity for lensing by these galaxies of
quasars (which mostly lie at higher redshift z 2).

The observed frequency of such lensed quasars is about what would be
expected in a standard = 1,
= 0 cosmology, so this data
sets fairly stringent upper limits:
0.70 at 90% C.L.
(Maoz & Rix 1993,
Kochanek 1993),
with more recent data giving even tighter constraints:
< 0.66 at 95% confidence
if m +
= 1
(Kochanek 1996).
This limit could
perhaps be weakened if there were (a) significant extinction by dust
in the E/S0 galaxies responsible for the lensing or (b) rapid
evolution of these galaxies, but there is much evidence that these
galaxies have little dust and have evolved only passively for z
1 (Steidel, Dickinson, &
Persson 1994;
Lilly et al. 1995;
Schade et al. 1996).
An alternative analysis by
Im, Griffiths, &
Ratnatunga (1997)
of some of the same optical lensing data considered by
Kochanek (1996)
leads them to deduce a value
= 0.640.26+0.15, which is barely consistent with
Kochanek's upper limit.
Malhotra, Rhodes, &
Turner (1997)
presents evidence for extinction of quasars by foreground galaxies
and claims that this weakens the lensing bound to
< 0.9, but this is not
justified quantitatively.
Maller, Flores, &
Primack (1997)
shows that edge-on disk galaxies can lens quasars very effectively, and
discusses a case in which optical extinction is significant. But the
radio observations discussed by
Falco, Kochanek, & Munoz
(1998),
which give a 2 limit
< 0.73, are not affected
by extinction. Recently
Chiba and Yoshii (1999)
have suggested that
a reanalysis of lensing using new models of the evolution of elliptical
galaxies gives =
0.7+0.1-0.2, but
Kochanek et al. (1999)
(see especially Fig. 4) show that the available evidence
disfavors the models of Chiba and Yoshii.

A model-dependent constraint appeared to come from simulations of
CDM
(Klypin, Primack, &
Holtzman 1996)
and OpenCDM
(Jenkins et al. 1998)
COBE-normalized models with h = 0.7,
m = 0.3, and
either = 0.7 or, for the open
case, = 0.
These models have too much power on small scales to be consistent with
observations, unless there is strong scale-dependent antibiasing of
galaxies with respect to dark matter. However, recent high-resolution
simulations
(Klypin et al. 1999)
find that merging and destruction of
galaxies in dense environments lead to exactly the sort of
scale-dependent antibiasing needed for agreement with observations for
the CDM model. Similar
results have been found using simulations plus semi-analytic methods
(Benson et al. 1999,
but cf.
Kauffmann et al. 1999).

Another constraint on from
simulations is a claim that the
number of long arcs in clusters is in accord with observations for an
open CDM model with
m = 0.3 but an order
of magnitude too low
in a CDM model with the same
m
(Bartelmann et al. 1998).
This apparently occurs because clusters with dense cores form too late
in such models. This is potentially a powerful constraint, and needs
to be checked and understood. It is now known that including cluster
galaxies does not alter these results
(Meneghetti et al. 1999;
Flores, Maller, &
Primack 1999).